Function-Valued Causal Influence in Nonlinear Time Series

Causal discovery in time series is increasingly performed using nonlinear machine-learning models, yet the resulting causal relationships are almost always summarized by scalar edge scores. We argue that this practice obscures the true object learned by nonlinear autoregressive models: a state-dependent function whose effect varies across regimes, magnitudes, and contexts. We formalize function-valued causal influence for additive, contribution-decomposable architectures and show that scalar causal scores constitute a severe information bottleneck, conflating between-state variation with within-state residual noise. Using Neural Additive Vector Autoregression as a representative architecture, we introduce a practical framework based on Individual Conditional Expectation for estimating causal response functions directly from trained models. Through controlled synthetic experiments, we demonstrate that edges with indistinguishable scalar scores can exhibit qualitatively different functional behaviors, including monotonic, thresholded, saturating, and sign-changing effects. An applied case study on democratic development further shows that function-valued analysis reveals regime-specific and asymmetric causal structure systematically missed by score-centric approaches.

When scientists use machine learning to study complex systems, for example, how democratic institutions influence each other over time, they typically get a single number for each relationship: a "causal score" that says how strongly one variable affects another. But this single number hides crucial information. A relationship that only activates above a critical threshold looks the same as one that operates uniformly across all conditions, even though the policy implications are completely different. We discovered that modern neural network models for time series actually learn rich, state-dependent causal functions, not simple numbers, but standard practice immediately throws away this detail by collapsing everything to a scalar score. We show mathematically that this creates a severe information bottleneck, and propose a practical method to recover the full functional picture from already-trained models without changing the model at all. Applying this to data on democratic institutions across 139 countries, we find that nearly every significant relationship between democratic components behaves differently depending on the political context: effects that appear uniformly positive in stable democracies reverse direction in more fragile ones. This matters because interventions designed to strengthen democracy based on scalar scores alone could be misguided. The real story only emerges when you look at the full functional relationship.